Theorem moan 2083

Description: "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moan	⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))

Proof of Theorem moan

Step	Hyp	Ref	Expression
1		simpr 109	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜑)
2	1	moimi 2079	1 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 103  ∃*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  moani  2084  mooran1  2086  mormo  2677  rmoan  2926